Which of the following is/are correct?

Options

A. The point of intersection of `vec(AB)` and `vec(CD)` where `A(4,7,8),B(-1,-2,1),C(2,3, 4)` and `D(1,2,5)` is `(3/2, 5/2, 9/2)`
B. If `(alpha, beta ,gamma)` be a point on the plane `2x+6y+15z=7` then the least value of `alpha^(2)+beta^(2)+25gamma^(2)` is
C. The circumcentre of the triangle formex by the points `(3,2,-5),(-3,8,-5)` and `(-3,2,1)` is the point `(-1,4,-3)`
D. The length of intercept on the line `(x+1)/(-1)=(y=12)/5=(z-7)/2` by the surface `11x^(2)-5y^(2)+z^(2)=0` is `sqrt(30)` units

Answer

A, C, D

Explanation

(A) Intersection point is mid -point of `AB` and `CD`
(B) `2alpha+6beta+15gamma=7`
`(2hati+6hatj+3hatk).(alpha hati+betahatj+5gammahatk) le 7xxsqrt(a^(2)+beta^(2)+25gamma^(2))`
`(2alpha+6beta+15gamma)le 7sqrt(alpha^(2)+beta^(2)+25gamma^(2))`
`alpha^(2)+beta^(2)+25gamma^(2)ge1`
(C) Triangle is equilateral
(D) Intersection points are `(1,2,3)` and `(2,-3,1)`